Elasticities of Demand for Non-Linear Equations

[achia]

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Part (a) has me a bit confused. To calculate elasticities, the general for would be (i) (%change in q)/(% change in p) = (d ln q) /(d ln p) (ii) (%change in q)/(% change in s) = (d ln q) /(d ln s)

Other than this, I am unsure of how to proceed.

I just need this part figured out since (c) depends on this. (c)-(e) is programming on GAUSS, which I have figured out. (b) is simple enough to do.

Any help would be appreciated!

 

[stapel]

For other users, here's the text from the image at the link:

22.4: Consider the following equation, where the quantity of wool demanded \(\displaystyle q\) depends on the price of wool \(\displaystyle p\) and the price of synthetics \(\displaystyle s\):

. . . . .\(\displaystyle q_1\, =\, \beta_1\, +\, \dfrac{\beta_2 \left(p_t^{\lambda}\, -\, 1\right)}{\lambda}\, +\, \dfrac{\beta_3 \left(s_t^{\lambda}\, -\, 1\right)}{\lambda}\, +\, e_t\). . .\(\displaystyle 22.5.1\)

where \(\displaystyle \beta_1,\, \beta_2,\, \beta_3,\) and \(\displaystyle \lambda \) are unknown parameters and \(\displaystyle e_t\) is an independent identically distributed random error with mean zero and variance \(\displaystyle \sigma^2\).

a) Find, in terms of the unknown parameters, the elasticity of demand for wool with respect to
. . .i) its own price
. . .ii) the price of synthetics

b) Show that Equation \(\displaystyle 22.5.1\) is
. . .i) a linear function of \(\displaystyle p\) and \(\displaystyle s\) if \(\displaystyle \lambda\, =\, 1\)
. . .ii) a linear function of \(\displaystyle \ln(p)\) and \(\displaystyle \ln(s)\) if \(\displaystyle \lambda\, =\, 0\)
. . .(Hint: Use l'Hospital's Rule to show that \(\displaystyle \lim_{\lambda \, \to\, \infty}\, \left(\dfrac{z^{\lambda}\, -\, 1}{\lambda}\right)\, =\, \ln(z)\))

c) Use the 45 observations given in Table 22.5 and nonlinear least squares to estimate the unknown parameters. Find corresponding elasticity estimates at the means of the sample data. Comment.

d) Test the hypotheses.
. . .i) \(\displaystyle \lambda\, =\, 0\)
. . .ii) \(\displaystyle \lambda\, =\, 1\)

e) Test the hypothesis that \(\displaystyle \beta_2\, =\, -\beta_3\)
Use nonlinear least squares...

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